# Properties

 Label 4800.2.a.f Level $4800$ Weight $2$ Character orbit 4800.a Self dual yes Analytic conductor $38.328$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.3281929702$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 4q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 4q^{7} + q^{9} + 4q^{11} - 2q^{13} + 6q^{17} - 4q^{19} + 4q^{21} - q^{27} - 2q^{29} - 4q^{31} - 4q^{33} - 2q^{37} + 2q^{39} + 2q^{41} - 4q^{43} + 8q^{47} + 9q^{49} - 6q^{51} + 10q^{53} + 4q^{57} - 4q^{59} - 6q^{61} - 4q^{63} - 4q^{67} + 16q^{71} + 6q^{73} - 16q^{77} - 4q^{79} + q^{81} - 12q^{83} + 2q^{87} + 10q^{89} + 8q^{91} + 4q^{93} + 14q^{97} + 4q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 0 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4800.2.a.f 1
4.b odd 2 1 4800.2.a.co 1
5.b even 2 1 192.2.a.c 1
5.c odd 4 2 4800.2.f.bh 2
8.b even 2 1 2400.2.a.r 1
8.d odd 2 1 2400.2.a.q 1
15.d odd 2 1 576.2.a.h 1
20.d odd 2 1 192.2.a.a 1
20.e even 4 2 4800.2.f.e 2
24.f even 2 1 7200.2.a.bx 1
24.h odd 2 1 7200.2.a.e 1
35.c odd 2 1 9408.2.a.bj 1
40.e odd 2 1 96.2.a.b yes 1
40.f even 2 1 96.2.a.a 1
40.i odd 4 2 2400.2.f.a 2
40.k even 4 2 2400.2.f.r 2
60.h even 2 1 576.2.a.g 1
80.k odd 4 2 768.2.d.h 2
80.q even 4 2 768.2.d.a 2
120.i odd 2 1 288.2.a.c 1
120.m even 2 1 288.2.a.b 1
120.q odd 4 2 7200.2.f.f 2
120.w even 4 2 7200.2.f.x 2
140.c even 2 1 9408.2.a.ct 1
240.t even 4 2 2304.2.d.s 2
240.bm odd 4 2 2304.2.d.c 2
280.c odd 2 1 4704.2.a.t 1
280.n even 2 1 4704.2.a.e 1
360.z odd 6 2 2592.2.i.h 2
360.bd even 6 2 2592.2.i.w 2
360.bh odd 6 2 2592.2.i.q 2
360.bk even 6 2 2592.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 40.f even 2 1
96.2.a.b yes 1 40.e odd 2 1
192.2.a.a 1 20.d odd 2 1
192.2.a.c 1 5.b even 2 1
288.2.a.b 1 120.m even 2 1
288.2.a.c 1 120.i odd 2 1
576.2.a.g 1 60.h even 2 1
576.2.a.h 1 15.d odd 2 1
768.2.d.a 2 80.q even 4 2
768.2.d.h 2 80.k odd 4 2
2304.2.d.c 2 240.bm odd 4 2
2304.2.d.s 2 240.t even 4 2
2400.2.a.q 1 8.d odd 2 1
2400.2.a.r 1 8.b even 2 1
2400.2.f.a 2 40.i odd 4 2
2400.2.f.r 2 40.k even 4 2
2592.2.i.b 2 360.bk even 6 2
2592.2.i.h 2 360.z odd 6 2
2592.2.i.q 2 360.bh odd 6 2
2592.2.i.w 2 360.bd even 6 2
4704.2.a.e 1 280.n even 2 1
4704.2.a.t 1 280.c odd 2 1
4800.2.a.f 1 1.a even 1 1 trivial
4800.2.a.co 1 4.b odd 2 1
4800.2.f.e 2 20.e even 4 2
4800.2.f.bh 2 5.c odd 4 2
7200.2.a.e 1 24.h odd 2 1
7200.2.a.bx 1 24.f even 2 1
7200.2.f.f 2 120.q odd 4 2
7200.2.f.x 2 120.w even 4 2
9408.2.a.bj 1 35.c odd 2 1
9408.2.a.ct 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4800))$$:

 $$T_{7} + 4$$ $$T_{11} - 4$$ $$T_{13} + 2$$ $$T_{19} + 4$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T$$
$5$ 1
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 - 6 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 2 T + 29 T^{2}$$
$31$ $$1 + 4 T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 - 2 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 - 10 T + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 + 6 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 - 16 T + 71 T^{2}$$
$73$ $$1 - 6 T + 73 T^{2}$$
$79$ $$1 + 4 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 - 10 T + 89 T^{2}$$
$97$ $$1 - 14 T + 97 T^{2}$$